Embedded newform 210.3.w.b.173.8

• Label: 210.3.w.b.173.8
• Level: $210$
• Weight: $3$
• Character: 210.173
• Analytic conductor: $5.722$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	210.w (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.72208555157\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			173.8
Character	\(\chi\)	\(=\)	210.173
Dual form			210.3.w.b.17.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(-0.489601 + 2.95978i) q^{3} +(1.73205 - 1.00000i) q^{4} +(1.47500 + 4.77749i) q^{5} +(0.414547 + 4.22234i) q^{6} +(6.99874 - 0.132847i) q^{7} +(2.00000 - 2.00000i) q^{8} +(-8.52058 - 2.89822i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{2} + 6 q^{3} + 12 q^{5} + 4 q^{7} + 128 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.36603	−	0.366025i	0.683013	−	0.183013i
							\(3\)	−0.489601	+	2.95978i	−0.163200	+	0.986593i
\(5\)	1.47500	+	4.77749i	0.295000	+	0.955497i
							\(7\)	6.99874	−	0.132847i	0.999820	−	0.0189781i
\(11\)	−9.30088	+	5.36987i	−0.845535	+	0.488170i								−0.859142	−	0.511738i	\(-0.829002\pi\)
							0.0136071	+	0.999907i	\(0.495669\pi\)
\(13\)	−3.28522	+	3.28522i	−0.252709	+	0.252709i	−0.822081	−	0.569371i	\(-0.807187\pi\)
							0.569371	+	0.822081i	\(0.307187\pi\)
\(17\)	16.0246	+	4.29378i	0.942624	+	0.252575i	0.697229	−	0.716848i	\(-0.254416\pi\)
							0.245394	+	0.969423i	\(0.421083\pi\)
\(19\)	−1.04608	+	1.81186i	−0.0550568	+	0.0953611i	−0.892240	−	0.451561i	\(-0.850867\pi\)
							0.837183	+	0.546922i	\(0.184201\pi\)
\(23\)	3.40405	+	12.7041i	0.148002	+	0.552351i	0.999603	+	0.0281612i	\(0.00896517\pi\)
							−0.851601	+	0.524190i	\(0.824368\pi\)
\(29\)	21.8689			0.754099			0.377049	−	0.926193i	\(-0.376939\pi\)
							0.377049	+	0.926193i	\(0.376939\pi\)
\(31\)	40.3894	−	23.3188i	1.30288	−	0.752221i	0.321987	−	0.946744i	\(-0.395649\pi\)
							0.980898	+	0.194523i	\(0.0623160\pi\)
\(37\)	−10.9244	−	40.7704i	−0.295254	−	1.10190i	−0.941016	−	0.338363i	\(-0.890127\pi\)
							0.645762	−	0.763539i	\(-0.276540\pi\)
\(41\)	−41.7957			−1.01941			−0.509703	−	0.860350i	\(-0.670245\pi\)
							−0.509703	+	0.860350i	\(0.670245\pi\)
\(43\)	−32.4714	−	32.4714i	−0.755148	−	0.755148i	0.220287	−	0.975435i	\(-0.429301\pi\)
							−0.975435	+	0.220287i	\(0.929301\pi\)
\(47\)	−18.0449	−	67.3444i	−0.383934	−	1.43286i	−0.839841	−	0.542833i	\(-0.817352\pi\)
							0.455907	−	0.890028i	\(-0.349315\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.w.b.173.8	yes	64
3.2	odd	2		210.3.w.a.173.11	yes	64
5.2	odd	4		210.3.w.a.47.16	yes	64
7.3	odd	6	inner	210.3.w.b.143.3	yes	64
15.2	even	4	inner	210.3.w.b.47.3	yes	64
21.17	even	6		210.3.w.a.143.16	yes	64
35.17	even	12		210.3.w.a.17.11	✓	64
105.17	odd	12	inner	210.3.w.b.17.8	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.11	✓	64	35.17	even	12
210.3.w.a.47.16	yes	64	5.2	odd	4
210.3.w.a.143.16	yes	64	21.17	even	6
210.3.w.a.173.11	yes	64	3.2	odd	2
210.3.w.b.17.8	yes	64	105.17	odd	12	inner
210.3.w.b.47.3	yes	64	15.2	even	4	inner
210.3.w.b.143.3	yes	64	7.3	odd	6	inner
210.3.w.b.173.8	yes	64	1.1	even	1	trivial